Recall the concepts of the class of metric (metric class) and the radiance obstruction of Mongé-Ampére manifolds $B$ with singularities. They are introduced in [KS04] and discussed in [GS06] in more details. We denote them by $k(B)\in H^{1}(B,i_{*}\tilde{\Lambda}^{\vee}\otimes\mathbb{R})$ and $c(B)\in H^{1}(B,i_{*}\Lambda)$, respectively. Here, $\Lambda$ is the affine structure as a $\mathbb{Z}^{\it dim(B)}$-local system in tangent bundle $T(B\setminus\Delta)$, $-^{\vee}$ denotes $-$’s dual local system, $\tilde{\Lambda}^{\vee}$ is local system of affine functions. In particular, we naturally have a morphism of local systems $f\colon\tilde{\Lambda}^{\vee}\to\Lambda^{\vee}$ which induces $f_{*}\colon H^{1}(B,i_{*}\tilde{\Lambda}^{\vee})\to H^{1}(B,i_{*}\Lambda^{\vee})$. It is also easy to see that, if we slightly change the definition of the metric class, to extract its “linear” part as $f_{*}k(B)$. Then, it naturally recovers the data $\overline{v}\in(e^{\perp}\otimes\mathbb{R}/\mathbb{R}e)$ i.e., we have $f_{*}k(\Phi_{\rm alg}([e,v]))=[v],$ under the natural identification $H^{1}(\Phi_{\rm alg}([e,v]),i_{*}\Lambda^{\vee}\otimes\mathbb{R})\hookrightarrow(e^{\perp}\otimes\mathbb{R}/\mathbb{R}e)$ which comes from the Leray spectral sequence applied to the elliptic fibration $X\twoheadrightarrow\Phi_{\rm alg}([e,v])$ in §4.2. Our results in [Od16] and Theorem 3.1 for $A_{g}$ can be re-interpretted similarly (but with weight $1$).